This is going to be one of the most abstract and technical posts yet; it’s inessential for reading this blog, but I wanted to include it because I would like to cultivate as many different possible viewpoints of scheme theory from the very start. There will be one more post running in this vein and then we will go back to doing actual geometry. However, this post will provide some more structure to the “functor of points” approach we have seen.

Essentially, when learning about the functor of points we saw that every scheme can be viewed as a set-valued presheaf on the opposite of the category of rings (i.e. just a functor from rings to sets, but this is a better viewpoint). But in fact this does not characterise schemes – not every presheaf on the opposite category of rings is the functor of points of a scheme (i.e. not every presheaf on the category of rings is representable). So in order to characterise schemes in this way we will need to find some further structure present. It turns out that with a suitable definition of a “topology” on the (opposite) category of rings – turning this category into something like a space – we will see that schemes are actually the sheaves on this category, which obey the “identity” and “glueing” axioms we looked at in one of the very first posts on this blog.

This post will consist of establishing the vocabulary of these so-called “Grothendieck (pre)topologies” on categories, and we will use these to define sheaves on categories in two ways. In an upcoming post we will use this to see the applicability to schemes, but this is going to be a purely category-theoretic post (and mostly taken from Johnstone’s lovely book Topos Theory). Feel free to skip!

**Pretopologies**

Let be a (small – objects form a set) category with pullbacks. A **Grothendieck pretopology** on is an assignment of a **set** to every object where the elements of are sets of -morphisms

into (called **covering families**) satisfying the following three axioms:

- For every , the family ;
- If is a morphism in and then ;
- If is an element of and is an element of for each then the composition is an element of .

Morally, these should be interpreted as follows: if is any topological space then (1) is an open covering of ; (2) if you’re given an open covering of a space and is a subset of then is an open covering of ; and (3) if you’re given an open covering and each of the open sets has an open covering then the collection of all the also covers .

As a concrete example, let be a topological space and let be the poset category of all its open subsets, ordered by inclusion (we saw this in this post). The pretopology on this category is just the set of open covers of each open set i.e. a covering family for is just a set of inclusions such that the union of all the equals .

Grothendieck pretopologies already give us enough “local” information in our categories to define a suitable notion of a **sheaf** on the category. Recall we could define a presheaf on any category as simply a contravariant functor, but in order to define a sheaf we needed to be able to talk about covers, and this is exactly what a pretopology lets us do. With this in mind we say that a presheaf on a category (taking values in sets) is a sheaf if for every covering family the diagram

is an equaliser. Here the first map is the universal map into the product induced by the restrictions for every morphism in the covering family. The top and bottom second maps are constructed as follows: let

be the canonical projections. For each we have a restriction map

Therefore the compositions induce a universal morphism . This is the first of the two maps. The second follows by first restricting to and then repeating the procedure. In general these will give different morphisms, but must agree when precomposed with the morphism from if is a sheaf.

We will use this definition of a sheaf sometimes because it is more closely related to what we have already seen. However it is a slightly vague definition; to see what I mean, suppose that satisfies this condition for a covering family . Then if is another family of covers containing , will also satisfy the sheaf condition with respect to . This means that two possibly different pretopologies can give rise to exactly the same sheaves.

This is a far-reaching quasi-generalisation of the fact that two different metrics on a space can induce exactly the same topology (the metrics then being called **equivalent**) and therefore the sheaves on these spaces (in the usual sense we defined here) will be exactly the same. Given that sheaves are more of a “topological” object and not a “metric” object, it might make sense to get to the root of this and define actual “topologies” on our categories rather than “metrics” (bear in mind this is a very loose analogy!). So we might want to consider covering families which are somehow “maximal”, and this is what the notion of a “sieve” encapsulates. Following this, we can also dispense with the assumption that our category has pullbacks; the drawback is that the notion of a sheaf becomes significantly more abstract.

**Topologies**

Let be a Grothendieck pretopology on a category , and let be an object. We say the set is a **sieve** on if

for every morphism .

**Definition**: Let be a small category. A **Grothendieck topology** on is a set of sieves on every object (called **covering sieves**) such that

- The
**maximal sieve**is an element of ; - If and is a morphism in then the
**pullback sieve**is an element of ; - If and is some sieve on such that for every morphism we have then .

A category equipped with a (Grothendieck) topology is called a **site**. This definition implies some important things:

Let and let be a sieve on containing i.e. every morphism in is in . Pick a morphism in ; then if is is any morphism in , the fact that is a sieve implies that , and therefore . But this shows that . Conversely it is clear that from the definitions; therefore . By axiom 3, it follows that .

Note that every Grothendieck pretopology induces a Grothendieck topology as follows: for each covering family , the collection of sieves on containing is a Grothendieck topology (check the axioms!). This is in some way (maybe) like how a metric induces a topology on a space, and in some sense this makes topologies more general.

The real benefit of using sieves instead of covering families is that we can drop the assumption about the base category having pullbacks. The reason is that we can always pull a sieve back along a morphism even if we cannot do this for the individual morphisms in the sieve. To see why this works, let be a sieve on an object and let

be the representable presheaf on induced by . Let

be the presheaf . Then is a presheaf on and is actually a **subpresheaf** of the representable presheaf . This means that in the functor category is a subobject of , but concretely it just means that for every , is a subset of . Now the presheaf category does have pullbacks (indeed, it has all limits and colimits and these are computed “pointwise”). By the Yoneda lemma we can think of as being fully and faithfully embedded in this presheaf category, so we can identify objects with their representable presheaves and sieves with the presheaves . This explains why we can dispense with having pullbacks, because they are present in this larger category in which sits.

**Sheaves on a site**

A **sheaf** on a site is then a presheaf on such that for every and every covering sieve , every morphism

in has exactly one extension to a morphism

Ouch! How abstract can you get???

But there’s a nice way to decode this to see it really isn’t too different to what we already have. Recall that by the Yoneda Lemma, natural transformations correspond bijectively to elements of the set . Now, a natural transformation consists of a function for every object that is compatible with morphisms i.e. we get the usual naturality squares for every morphism.

If is not the domain of any of the morphisms in the sieve then . In this case the function is the unique “empty function”. If not, produces an element of for every element . Furthermore, if is a morphism then since is a sieve, is nonempty and we obtain elements

for each . Mapping these into via we see that equals .

Therefore a morphism amounts to specifying a collection of “sections” in every for each arising as a domain of one of the elements in which are “compatible” in the sense that the “restriction maps” induced by morphisms take sections to sections. By the definition above, being a sheaf means there is a unique natural transformation extending . But since this corresponds bijectively to an element of , this means there is a unique section such that the “restriction” of to each (by which I mean the map induced by some ) equals . And this looks a lot more like our definition of a sheaf we originally gave!

Here are some final notes for this post:

- In the last paragraph we saw how unique “sections” can be glued up uniquely from smaller sections when is a sheaf. The crucial difference (which confused me for a while) is that whereas in the topological case there was always at most one map (corresponding to the inclusion of an open subset), this time there may be many different ways to “restrict a section”; in fact we can do this along each of the different morphisms . So in there may be many different sections , s_\beta, \dots $ all being “restrictions” of the same “global section” ; this is fine, just noting that they have been “restricted” in different ways.
- Sometimes it is more useful to work with a topology and other times a pretopology may be better suited. I get the feeling from reading the experts that pretopologies may be more suitable for actually computing things, whereas topologies and sites are more elegant, but really I’m not sure. I’m only beginning to get my head around these things.

Finally, in one of the next few posts I will attempt to use these gadgets to show how every scheme, as we’ve defined them, can also be defined as a **sheaf on the Zariski site**.

Just found your link on Brian Conrad’s IUT blog posting. I’m a physicist very interested in these categorical ideas of “generalized space” and will be reading your blog posts to help me understand things. Haven’t had a chance to read much yet, but thanks for writing these things up in a more accessible way than the math texts I’ve encountered thus far!

LikeLike

Thank you for your support Doug! I’m really glad to hear that this has also been interesting from a physicist’s perspective; in what sort of areas in physics are these “topos-theoretic” ideas used?

LikeLike